ICSE Class 10 Maths Chapter 08 Ratio

Chapter 8

Ratio

Points To Remember

1. Ratio: If a and b are two non-zero quantities of the same kind and in the same units, then the fraction \(\frac{a}{b}\) is called the ratio between a and b, and is written as a : b. a and b are called its terms, a is the first term or antecedent and b, the second term or consequent.

2. An Important Property: The value of a ratio remains unchanged, if both of its terms are multiplied or divided by the same non-zero number.

Proof: Let a : b be given ratio and let m be a non-zero number.

Then, we may write, a : b as \(\frac{a}{b}\).

(i) Clearly, we have \(\frac{a}{b} = \frac{ma}{mb}\). This shows that a : b is the same as ma : mb.

(ii) We may write, \(\frac{a}{b} = \frac{(a/m)}{(b/m)}\). This shows that a : b is the same as \(\frac{a}{m} : \frac{b}{m}\).

3. A Ratio in its Lowest Terms: When a and b have no common factor, other than 1, we say that a : b is in its lowest terms.

4. Composition of Ratios:

(i) Duplicate ratio of a : b is \(a^2 : b^2\).

(ii) Triplicate ratio of a : b is \(a^3 : b^3\).

(iii) Sub-duplicate ratio of a : b is \(\sqrt{a} : \sqrt{b}\), or \(a^{\frac{1}{2}} : b^{\frac{1}{2}}\).

(iv) Sub-triplicate ratio of a : b is \(a^{1/3} : b^{1/3}\) or \(\sqrt[3]{a} : \sqrt[3]{b}\).

(v) Reciprocal ratio of a : b is \(\frac{1}{a} : \frac{1}{b}\), which is same as b : a.

(vi) Compounded ratio of a : b and c : d is ac : bd.

5. Proportion: Four non-zero quantities a, b, c, d are said to be in proportion, if a : b = c : d and we write, a : b : : c : d. We read it as, "a is to b as c is to d".

If a : b : : c : d, then:

(i) a, b, c, d are known as first, second, third and fourth term respectively.

Teacher's Note

Ratios help us compare quantities in everyday life - like when following a recipe that requires ingredients in specific proportions, or when adjusting a map scale to understand real distances.

(ii) d is called the fourth proportional to a, b, c.

(iii) a and d are called extremes, while b and c are called means.

An Important Property

If a : b : : c : d, then the product of extremes is equal to the product of means.

Proof: a : b : : c : d \(\Leftrightarrow \frac{a}{b} = \frac{c}{d}\)

\(\Leftrightarrow ad = bc\)

Therefore, Product of extremes = product of means.

6. Continued Proportion

We say that a, b, c, d, e, f etc. are in continued proportion, if

\(\frac{a}{b} = \frac{b}{c} = \frac{c}{d} = \frac{d}{e} = \frac{e}{f}\).

7. Mean Proportional (or Geometric Mean)

Let a, b, c be in continued proportion.

Then, \(\frac{a}{b} = \frac{b}{c}\) or \(b^2 = ac\) or \(b = \sqrt{ac}\).

Here, b is called the mean proportional or geometric mean between a and c.

8. Third Proportional:

If a : b = b : c, then c is called the third proportional to a and b. Clearly, it is the fourth proportional to a, b, b.

9. An Important Property:

If \(\frac{a}{b} = \frac{c}{d} = \frac{e}{f}\), prove that each ratio in equality \(\frac{a + c + e}{b + d + f}\).

Proof: Let \(\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = k\),

then a = bk, c = dk and e = fk.

\(\frac{a + c + e}{b + d + f} = \frac{bk + dk + fk}{b + d + f}\)

\(= \frac{k(b + d + f)}{(b + d + f)} = k\).

Hence, each ratio = \(\frac{a + c + e}{b + d + f}\)

= Sum of antecedent / Sum of consequents

10. Properties of Ratio and Proportion:

(1) Invertendo: If a : b : : c : d, then b : a : : d : c.

Proof: a : b : : c : d \(\Rightarrow \frac{a}{b} = \frac{c}{d} \Rightarrow \frac{b}{d} = \frac{d}{c}\) [Therefore, Reciprocals of equals are equal]

Teacher's Note

Understanding these properties helps in solving real problems like distributing resources fairly among people or adjusting quantities in science experiments.

\(\Rightarrow\) b : a : : d : c.

Hence, a : b : : c : d \(\Rightarrow\) b : a : : d : c. [Invertendo]

(2) Alternendo: If a : b : : c : d, then a : c : : b : d.

Proof: a : b : : c : d \(\Rightarrow \frac{a}{b} = \frac{c}{d}\)

\(\Rightarrow ad = bc\) [By cross multiplication]

\(\Rightarrow \frac{a}{c} = \frac{b}{d}\) [On dividing both sides by cd]

\(\Rightarrow a : c : : b : d\)

Hence, a : b : : c : d \(\Rightarrow\) a : c : : b : d [Alternendo]

(3) Componendo: If a : b : : c : d, then (a + b) : b : : (c + d) : d.

Proof: a : b : : c : d \(\Rightarrow \frac{a}{b} = \frac{c}{d}\)

\(\Rightarrow \frac{a}{b} + 1 = \frac{c}{d} + 1\) [Adding 1 on both sides]

\(\Rightarrow \frac{a + b}{b} = \frac{c + d}{d}\)

\(\Rightarrow (a + b) : b : : (c + d) : d\).

Hence, a : b : : c : d \(\Rightarrow\) (a + b) : b : : (c + d) : d [componendo]

(4) Dividendo: If a : b : : c : d, then (a - b) : b : : (c - d) : d.

Proof: a : b : : c : d \(\Rightarrow \frac{a}{b} = \frac{c}{d}\)

\(\Rightarrow \frac{a}{b} - 1 = \frac{c}{d} - 1\) [On subtracting 1 from both sides]

\(\Rightarrow \frac{a - b}{b} = \frac{c - d}{d}\)

\(\Rightarrow (a - b) : b : : (c - d) : d\).

Hence, a : b : : c : d \(\Rightarrow\) (a - b) : b : : (c - d) : d [Dividendo]

(5) Componendo & Dividendo: If a : b : : c : d, then (a + b) : (a - b) : : (c + d) : (c - d).

Proof: a : b : : c : d.

\(\Rightarrow \frac{a}{b} = \frac{c}{d}\)

\(\Rightarrow \frac{a + b}{b} = \frac{c + d}{d}\) ....(i) [By componendo]

and \(\frac{a - b}{b} = \frac{c - d}{d}\) ....(ii) [By dividendo]

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